Past Papers’ Solutions | Cambridge International Examinations (CIE) | AS & A level | Mathematics 9709 | Pure Mathematics 1 (P1-9709/01) | Year 2018 | Feb-Mar | (P1-9709/12) | Q#8

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Question

A curve has equation .

i.Find the x-coordinates of the stationary points.

ii.Find .

iii.Find, showing all necessary working, the nature of each stationary point.

Solution

i.

We are required to find the coordinates of the stationary point of the curve.

Coordinates of stationary point on the curve can be found from the derivative of equation of the  curve by equating it with ZERO. This results in value of x-coordinate of the stationary point on the curve. 

Therefore, first we find the derivative of equation of the curve.

Gradient (slope) of the curve is the derivative of equation of the curve. Hence gradient of curve with respect to  is:

We are given equation of the curve as;

Therefore;

Rule for differentiation of  is:

Rule for differentiation of  is:

Now, to find the coordinates of stationary point of the curve;

Let ; then , therefore;

Now we have two options.

Since


ii.

Second derivative is the derivative of the derivative. If we have derivative of the curve   as , then  expression for the second derivative of the curve  is;

We have found in (i) that;

Therefore;

Rule for differentiation of  is:

Rule for differentiation of  is:

Rule for differentiation of  is:


iii.

Next we are required to find the nature of each stationary point.

Once we have the x-coordinate of the stationary point of a curve, we can determine its  nature, whether minimum or maximum, by finding 2nd derivative of the curve.

We substitute of the stationary point in the expression of 2nd derivative of the curve and  evaluate it;

If  or then stationary point (or its value) is minimum.

If  or then stationary point (or its value) is maximum.

From (i) we have x-coordinates of s a stationary points on the curve as 16 & 4.

Therefore, substituting  in expression of second derivative;

Since , the stationary point at  is a minimum.

Next, substituting  in expression of second derivative;

Since , the stationary point at  is a maximum.

 

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